DOCSLIB.ORG

From Metric Properties to a Rigidity Conjecture on Finsler Manifolds Via a Geodesic Detecting Algorithm

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
From Metric Properties to a Rigidity Conjecture on Finsler Manifolds Via a Geodesic Detecting Algorithm From Metric Properties to a Rigidity Conjecture on Finsler Manifolds via a Geodesic Detecting Algorithm Alexandru Kristály and Ágoston Róth Abstract. The aim of this paper is to consider Busemann-type inequalities on Finsler manifolds. We actually formulate a rigidity conjecture: any Finsler manifold which is a Busemann NPC space is Berwaldian. This statement is supported by some theoretical results and numerical examples. The presented examples are ob- tained by using evolutionary techniques (genetic algorithms) which can be used for detecting geodesics on a large class of not necessarily reversible Finsler manifolds. 2000 Mathematics Subject Classi…cation: 58B20, 53C60, 53C70, 32Q05. Keywords: Busemann NPC space, Finsler manifold, Berwald space, numerical detection of geodesics via genetic algorithms. 1 Introduction In the forties, Busemann developed a synthetic geometry of G-spaces, see [3]. These spaces possess the essential geometric properties of Finsler manifolds but they have no necessarily a di¤erential structure. Nowadays, Busemann’scurvature is referred to non-positively curved (quasi-)metric spaces, called Busemann NPC spaces; this notion of non-positive curvature requires that in small geodesic triangles the length of a side is at least the twice of the geodesic distance of the mid-points of the other two sides, see [3, p. 237]. Note that on Riemannian manifolds the non-positivity of the sectional curvature characterizes Busemann’snon-positivity curvature. In the …fties, Aleksandrov introduced independently another notion of curvature on metric spaces, a stronger one than that of Busemann, see Jost [5, Section 2.3]. Nevertheless, on Riemannian manifolds the Aleksandrov curvature condition holds if and only if the sectional curvature is non-positive, see Bridson-Hae‡iger[2, Theorem 1A.6]. In the Finsler case the picture is quite rigid: if on a reversible Finsler manifold (M; F ) the Aleksandrov curvature condition holds (on the induced metric space by (M; F )) then (M; F ) should be Riemannian, see [2, Proposition 1.14]. Having these facts in our mind, a natural question arises: What about Finsler manifolds that are Busemann NPC spaces? Clearly, we are thinking …rst to those Finsler manifolds which have non-positive ‡ag curvature. However, within this class, we easily …nd examples which are not Busemann NPC spaces. For instance, the Hilbert metric of the interior of a simple, closed and strictly convex curve c in the Euclidean plane de…nes a Finsler metric with constant ‡ag curvature 1, but Busemann’s curvature condition holds if and only if c is an ellipse, see Kelly-Straus [6]. In the latter case, the Hilbert metric becomes a Riemannian one. The above example suggests we face again a rigidity result as in the Aleksandrov case. However, a recent result of Kozma-Kristály-Varga [8] entitle us to believe that the appropriate class of Finsler manifolds is the Berwald one, a larger class than Riemannian manifolds. Namely, we formulate the following Conjecture. Let (M; F ) be a (not necessarily reversible) Finsler manifold such that (M; dF ) is a Busemann NPC space. Then (M; F ) is a Berwald space. The aim of the present paper is to recall some theoretical developments and certain numerical examples in order to support the above Conjecture. In addition, we propose a new numerical approach to detect minimal geodesics on Finsler manifolds by using evolutionary techniques. Note that the presented numerical approach can be e¢ ciently used for verifying di¤erent metric relations on a large class of Finsler manifolds exploiting the basic property of Finslerian-geodesics that they locally minimize the integral length. The paper is organized as follows. In the next section we recall the notion of Busemann NPC (quasi-)metric spaces and basic elements of Finsler manifolds. Section 3 deals with the Conjecture from both theoretical and numerical point of view. Section 4 is independent from the previous ones and describes the detection of geodesics on Finsler manifolds by using genetic algorithms. 1 2 Preliminaries 2.1 Busemann NPC spaces Let (M; d) be a quasi-metric space (i.e., d is not necessarily symmetric). A continuous curve c : [0; r] M with c (0) = p; c (r) = q is a shortest geodesic, if l (c) = d (p; q), where l (c) is the generalized length of !c and it is de…ned by n l (c) = sup d (c (ti 1) ; c (ti)) : 0 = t0 < t1 < ::: < tn = r; n N : 2 (i=1 ) X We say that (M; d) is a locally geodesic (length) space if for every point p M there is a neighborhood 2 Np M such that for every two points x; y Np there exists a shortest geodesic joining them. In the sequel, we always assume that the shortest geodesics2 are parametrized proportionally to the arc length, i.e., l c = t l (c) ; t [0; r] : j[0;t] 8 2 De…nition 1 (Busemann NPC space) A locally geodesic space (M; d) is said to be a Busemann non-positive curvature space (shortly, Busemann NPC space), if for every p M there exists a neighborhood Np M 2 such that for any two shortest geodesics c1; c2 : [0; 1] M with c1 (0) = c2 (0) Np and with endpoints ! 2 c1 (1) ; c2 (1) Np we have 2 1 1 2d c1 ; c1 d(c1 (1) ; c2 (1)): (1) 2 2 2.2 Finsler manifolds Let M be a connected m-dimensional C1 manifold and let TM = p M TpM be its tangent bundle. If the 2 continuous function F : TM [0; ) satis…es the conditions that it is C1 on TM 0 ; F (p;ty) = tF (p; y) for ! 1 S nf g 1 2 all t 0 and y TpM; i.e., F is positively homogeneous of degree one; and the matrix gij(y) := F (y) 2 2 yiyj is positive de…nite for all y TM 0 ; then we say that (M; F ) is a Finsler manifold. If F is absolutely homogeneous, then (M; F ) is2 said ton be f greversible. Let TM be the pull-back of the tangent bundle TM by : TM 0 M: Unlike the Levi-Civita connection in Riemann geometry, there is no unique natural connectionn in f g the ! Finsler case. Among these k connections on T M; we choose the Chern connection whose coe¢ cients are denoted by ij; see Bao-Chern- Shen [1, p. 38]. This connection induces the curvature tensor, denoted by R; see [1, Chapter 3]. The Chern connection de…nes the covariant derivative DVU of a vector …eld U in the direction V TpM: Since, in i 2 general, the Chern connection coe¢ cients jk in natural coordinates have a directional dependence, we must say explicitly that DVU is de…ned with a …xed reference vector. In particular, let c : [0; r] M be a smooth k i ! curve with velocity …eld T = T(t) = c_(t) = T k=1;m : Suppose that U = U i=1;m and W are vector …elds de…ned along c: We de…ne D U with reference vector W as T i dU j k i @ DTU = + U T (jk)(c;W) i ; dt @p c(t) j @ where i is a basis of Tc(t)M: A C1 curve c : [0; r] M; with velocity T = c_ is a (Finslerian) @p c(t) j i=1;m ! geodesicn if o T D = 0 with reference vector T. (2) T F (T) If the Finslerian speed of the geodesic c is constant, then (2) becomes d2ci dcj dck + i = 0; i = 1; :::; m = dim M: (3) dt2 dt dt jk (c;T) In the sequel, we assume that the geodesics have constant Finslerian speed. 2 A Finsler manifold (M; F ) is said to be forward (resp. backward) geodesically complete if every geodesic c : [0; 1] M can be extended to a geodesic de…ned on [0; ) (resp. ( ; 1]). ! 1 1 Let c : [0; r] M be a piecewise C1 curve. Its integral length is de…ned as ! r LF (c) = F (c (t) ; c_ (t)) dt: Z0 For p; q M, denote by (p; q) the set of all piecewise C1 curves c : [0; r] M such that c (0) = p and 2 [0;r] ! c (r) = q. De…ne the map dF : M M [0; ) by ! 1 dF (p; q) = inf LF (c) : (4) c (p;q) 2 [0;r] Of course, we have dF (p; q) 0; where equality holds if and only if p = q; and the triangle inequality holds, i.e., dF (p0; p2) dF (p0; p1) + dF (p1; p2) for every p0; p1; p2 M: In general, since F is only a 2 positive homogeneous function, dF (p; q) = dF (q; p); thus, (M; dF ) is only a quasi-metric space. If (M; g) is a 6 Riemannian manifold, we will use the notation dg instead of dF which becomes a usual metric function. Busemann and Mayer [4, Theorem 2, p. 186] proved that the generalized length l(c) and the integral length LF (c) of any (piecewise) C1 curve coincide for Finsler manifolds. Let (p; y) TM 0 and let V be a section of the pulled-back bundle TM. Then, 2 n f g g(p;y) (R (V; y) y; V) K (y; V) = 2 (5) g (y; y) g (V; V) g (y; V) (p;y) (p;y) (p;y) is the ‡ag curvature with ‡ag y and transverse edge V. Here, i j 1 2 i j g(p;y) := gij(p;y)dp dp := F dp dp ; p M; y TpM; (6) 2 i j 2 2 y y is the Riemannian metric on the pulled-back bundle TM.
Load more

Recommended publications
  • Riemannian and Finslerian Geometry for Diffusion Weighted Magnetic Resonance Imaging
    Riemannian and Finslerian geometry for diffusion weighted magnetic resonance imaging Luc Florack Computational Brain Connectivity Mapping Winter School Workshop 2017 - November 20-24, Jeans-les-Pins, France © Nicolas Lavarenne, l’exposition ‘A ciel ouvert’ Heuristics Finsler manifold. A Finsler manifold is a space (M,F) of spatial base points x∈M, furnished with a notion of a ‘line’ or ‘length element’ ds ≐ F(x,dx). The ‘infinitesimal displacement vectors’ dx are ‘infinitely scalable’ into finite ‘tangent’ or ‘velocity vectors’ ẋ, viz. dx = ẋ dt. The collection of all (x, ẋ) is called the tangent bundle TM over M. Integrating the line element along a curve C produces the ‘length’ of that curve: L (C)= ds = F (x, dx) ZC ZC Bernhard Riemann Sophus Lie Elie Cartan Albert Einstein Paul Finsler Applications Examples (anisotropic media). Mechanics e.g. F(x,dx) := infinitesimal displacement, or infinitesimal travel time, etc. Optimal control e.g. F(x,dx) := local cost function for infinitesimal movement of Reeds-Shepp car Optics e.g. F(x,dx) := infinitesimal travel time for light propagation Seismology e.g. F(x,dx) := infinitesimal travel time for seismic ray propagation Ecology e.g. F(x,dx) := infinitesimal energy for coral reef state transition Relativity e.g. F(x,dx) := infinitesimal (pseudo-Finslerian) spacetime line element Diffusion MRI e.g. F(x,dx) := infinitesimal hydrogen spin diffusion Axiomatics Literature. © David Bao et al. © Hanno Rund et al. The Riemann-DTI paradigm DWMRI signal attenuation. E(x, q, ⌧)=exp[ ⌧D(x, q, ⌧)] − q = γ¯ g(t)dt q-space variable Z 2⇡iq ⇠ Propagator.
    [Show full text]
  • Heat Flow on Finsler Manifolds
    Heat Flow on Finsler Manifolds Shin-ichi Ohta,∗ Karl-Theodor Sturm August 4, 2008 This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F (x, ) : T M R on each tangent space. Mostly, we · x → + will require that this norm is strongly convex and smooth and that it depends smoothly on the base point x. The particular case of a Hilbert norm on each tangent space leads to the important subclasses of Riemannian manifolds where the heat flow is widely studied and well understood. We present two approaches to the heat flow on a Finsler manifold: either as gradient flow on L2(M,m) for the energy • 1 (u)= F 2( u) dm; E 2 ∇ ZM or as gradient flow on the reverse L2-Wasserstein space (M) of probability measures on • P2 M for the relative entropy Ent(u)= u log u dm. ZM Both approaches depend on the choice of a measure m on M and then lead to the same nonlinear evolution semigroup. We prove 1,α-regularity for solutions to the (nonlinear) heat equation on C the Finsler space (M,F,m). Typically, solutions to the heat equation will not be 2. More- C over, we derive pointwise comparison results ´ala Cheeger-Yau and integrated upper Gaussian estimates ´ala Davies. 1 Finsler manifolds 1.1 Finsler structures Throughout this paper, a Finsler manifold will be a pair (M, F ) where M is a smooth, connected n-dimensional manifold and F : T M R is a measurable function (called Finsler structure) → + with the following properties: (i) F (x,cξ)= cF (x,ξ) for all (x,ξ) T M and all c> 0.
    [Show full text]
  • On the Existence of Convex Functions on Finsler Manifolds
    On the existence of convex functions on Finsler manifolds ∗† Sorin V. SABAU and Pattrawut CHANSANGIAM Abstract We show that a non-compact (forward) complete Finsler manifold whose Holmes- Thompson volume is finite admits no non-trivial convex functions. We apply this result to some Finsler manifolds whose Busemann function is convex. 1 Introduction Finsler manifolds are a natural generalization of Riemannian ones in the sense that the metric depends not only on the point, but on the direction as well. This generalization implies the non-reversibility of geodesics, the difficulty of defining angles and many other particular features that distinguish them from Riemannian manifolds. Even though classi- cal Finsler geometry was mainly concerned with the local aspects of the theory, recently a great deal of effort was made to obtain global results in the geometry of Finsler manifolds ([3], [13], [15], [17] and many others). In a previous paper [16], by extending the results in [9], we have studied the ge- ometry and topology of Finsler manifolds that admit convex functions, showing that such manifolds are subject to some topological restrictions. We recall that a func- tion f : (M, F ) R, defined on a (forward) complete Finsler manifold (M, F ), is → called convex if and only if along every geodesic γ :[a, b] M, the composed func- tion ϕ := f γ :[a, b] R is convex, that is → ◦ → f γ[(1 λ)a + λb] (1 λ)f γ(a)+ λf γ(b), 0 λ 1. (1.1) ◦ − ≤ − ◦ ◦ ≤ ≤ arXiv:1811.02121v2 [math.DG] 6 Jan 2019 If the above inequality is strict for all geodesics γ, the function f is called strictly convex, and if the equality holds good for all geodesics γ, then f is called linear.
    [Show full text]
  • Sasakian Finsler Structures on Diffusion of Tangent Bundles Satisfying 푪∗ 푿푯, 풀푯 푺 = ퟎ
    International Journal of Engineering Technology Science and Research IJETSR www.ijetsr.com ISSN 2394 – 3386 Volume 3, Issue 10 October 2016 Sasakian Finsler Structures on Diffusion of Tangent Bundles ∗ 푯 푯 Satisfying 푪 푿 , 풀 푺 = ퟎ Nesrin Caliskan1,, A. Funda Yaliniz2 1Faculty of Education, Department of Elementary Mathematics Education, Usak University, 64200, Usak- TURKEY 2Department of Mathematics, Faculty of Art and Sciences, Dumlupinar University, 43100, Kutahya-TURKEY Abstract. Inthis paper,the quasi-conformal curvature tensor 퐶∗ of Sasakian Finsler structures ontangent bundlesis considered. The type of these manifolds satisfying퐶∗ 푋퐻, 푌퐻 푆 = 0 are discussed where푆 denotes the Ricci tensor field 0 0 of type and 푋퐻, 푌퐻휖 푇 푇푀 퐻 . In addition, relations between these structures and 휂-Einstein structures are 2 2 푢 0 examined. Mathematics Subject Classification (2010). 53D15, 53C05, 53C15, 53C60. Keywords.Quasi-conformal curvature tensor; 휼-Einstein;tangent bundle; Sasakian Finsler structure. 1. Introduction and main results There are many different frameworks to study Finsler geometry (like in [1,2]).In this study, we contextualized by using double tangent bundles of Finsler manifolds (such as [3,4,5]). Quasi-conformal curvature tensor is studied in papers for several structures(like[2,6-9,10]). In this study, the notion of “quasi-conformal curvature tensor” for Sasakian Finsler Structures on diffusions of tangent bundles is defined and we found different invariants. In order to study this, essential characteristics of these structures are given; Let 푀 be an 푚 = 2푛 + 1 -dimensional smooth manifold and푇푀 its tangent bundle, 푢 = 푥, 푦 ∈ 푇푀. Ifa function퐹: 푇푀 → 0, ∞ satisfies 1 ∞ i) 퐹 ∈ 퐶 (푇푀) and 퐹 ∈ 퐶 (푇푀0), 푖 푖 푖 푖 ii) 퐹 푥 , 휆푦 = 휆퐹 푥 , 푦 , 휆 > 0, 1 휕2퐹2 iii) The 푚 × 푚 Hessian matrix 푔 푥, 푦 = is positive definite on 푇푀 , 푖푗 2 휕푦 푖휕푦 푗 0 푚 then퐹 = 푀, 퐹 is called a Finsler manifold and푔 is a Finsler metric tensor with 푔푖푗 coefficients [11].
    [Show full text]
  • Landsberg Curvature, S-Curvature and Riemann Curvature
    Riemann{Finsler Geometry MSRI Publications Volume 50, 2004 Landsberg Curvature, S-Curvature and Riemann Curvature ZHONGMIN SHEN Contents 1. Introduction 303 2. Finsler Metrics 304 3. Cartan Torsion and Matsumoto Torsion 311 4. Geodesics and Sprays 313 5. Berwald Metrics 315 6. Gradient, Divergence and Laplacian 317 7. S-Curvature 319 8. Landsberg Curvature 321 9. Randers Metrics with Isotropic S-Curvature 323 10. Randers Metrics with Relatively Isotropic L-Curvature 325 11. Riemann Curvature 327 12. Projectively Flat Metrics 330 13. The Chern Connection and Some Identities 333 14. Nonpositively Curved Finsler Manifolds 337 15. Flag Curvature and Isotropic S-Curvature 341 16. Projectively Flat Metrics with Isotropic S-Curvature 342 17. Flag Curvature and Relatively Isotropic L-Curvature 348 References 351 1. Introduction Roughly speaking, Finsler metrics on a manifold are regular, but not neces- sarily reversible, distance functions. In 1854, B. Riemann attempted to study a special class of Finsler metrics | Riemannian metrics | and introduced what is now called the Riemann curvature. This infinitesimal quantity faithfully reveals the local geometry of a Riemannian manifold and becomes the central concept of Riemannian geometry. It is a natural problem to understand general regu- lar distance functions by introducing suitable infinitesimal quantities. For more than half a century, there had been no essential progress until P. Finsler studied the variational problem in a Finsler manifold. However, it was L. Berwald who 303 304 ZHONGMIN SHEN first successfully extended the notion of Riemann curvature to Finsler metrics by introducing what is now called the Berwald connection. He also introduced some non-Riemannian quantities via his connection [Berwald 1926; 1928].
    [Show full text]
  • Arxiv:1807.08398V1 [Math.DG] 23 Jul 2018 Set by Ifold Usin1.1
    ON FINSLER TRANSNORMAL FUNCTIONS MARCOS M. ALEXANDRINO, BENIGNO O. ALVES, AND HENGAMEH R. DEHKORDI Abstract. In this note we discuss a few properties of transnormal Finsler functions, i.e., the natural generalization of distance functions and isopara- metric Finsler functions. In particular, we prove that critical level sets of an analytic transnormal function are submanifolds, and the partition of M into level sets is a Finsler partition, when the function is defined on a compact analytic manifold M. 1. Introduction Let (M, F ) be a forward complete Finsler manifold. A function f : M R is called F -transnormal function if F ( f)2 = b(f) for some continuous function→ b. Transnormal functions on Riemannian∇ manifolds have been focus of researchers in the last decades. In particular, if b C2(f(M)) then the level sets of f are leaves of the so called singular Riemannian∈ foliation and the regular level sets are equifocal hypersurfaces; see e.g, [14], [15] and [2, Chapter 5]. In Finsler geometry, the study of transnormal functions has just begun, see [6] but there are already some interesting applications in wildfire modeling, see [11]. The most natural example of a transnormal function on a Finsler space is the dis- tance function on a Minkowski space. More precisely consider a Randers Minkowski space (V, Z) and define f(x) := d(0, x). It is well known that in this example b = 1, see [16, Lemma 3.2.3]. And already here one can see a phenomenon that does not −1 exist in the Riemannian case. The regular level set f (c1) is forward parallel to −1 −1 −1 f (c2) if c1 <c2 but f (c2) is not forward parallel to f (c1) and hence the par- −1 tition = f (c) c∈(0,∞) is not a Finsler partition of V 0, recall basic definitions and examplesF { in Sections} 2 and 3 respectively.
    [Show full text]
  • The Geometry of a Randers Rotational Surface with an Arbitrary Direction Wind
    mathematics Article The Geometry of a Randers Rotational Surface with an Arbitrary Direction Wind Rattanasak Hama 1,*,† and Sorin V. Sabau 2,† 1 Faculty of Science and Industrial Technology, Surat Thani Campus, Prince of Songkla University, Surat Thani 84000, Thailand 2 Department of Biological Sciences, Sapporo Campus, Tokai University, Sapporo 005-8601, Japan; [email protected] * Correspondence: [email protected] † These authors contributed equally to this work. Received: 5 October 2020; Accepted: 11 November 2020; Published: 17 November 2020 Abstract: In the present paper, we study the global behaviour of geodesics of a Randers metric, defined on Finsler surfaces of revolution, obtained as the solution of the Zermelo’s navigation problem. Our wind is not necessarily a Killing field. We apply our findings to the case of the topological cylinder R × S1 and describe in detail the geodesics behaviour, the conjugate and cut loci. Keywords: geodesics; cut locus; Finsler manifolds 1. Introduction A Finsler surface (M, F) is a geometrical structure defined on a smooth 3-manifold S ⊂ TM whose the canonical projection p : S ! M is a surjective submersion such that for each x 2 M, the p-fiber −1 Sx = p (x) is a strictly convex curve including the origin Ox 2 Tx M. Here, we denote by TM the tangent bundle of M. This is actually equivalent to saying that (M, F) is a surface M endowed with a Minkowski norm in each tangent space Tx M that varies smoothly with the base point x 2 M all over the manifold. Obviously, S is the unit sphere bundle f(x, y) 2 TM : F(x, y) = 1g, also called the indicatrix bundle.
    [Show full text]
  • Finsler Metric, Systolic Inequality, Isometr
    DISCRETE SURFACES WITH LENGTH AND AREA AND MINIMAL FILLINGS OF THE CIRCLE MARCOS COSSARINI Abstract. We propose to imagine that every Riemannian metric on a surface is discrete at the small scale, made of curves called walls. The length of a curve is its number of wall crossings, and the area of the surface is the number of crossings of the walls themselves. We show how to approximate a Riemannian (or self-reverse Finsler) metric by a wallsystem. This work is motivated by Gromov's filling area conjecture (FAC) that the hemisphere minimizes area among orientable Riemannian sur- faces that fill a circle isometrically. We introduce a discrete FAC: every square-celled surface that fills isometrically a 2n-cycle graph has at least n(n−1) 2 squares. We prove that our discrete FAC is equivalent to the FAC for surfaces with self-reverse metric. If the surface is a disk, the discrete FAC follows from Steinitz's algo- rithm for transforming curves into pseudolines. This gives a combinato- rial proof of the FAC for disks with self-reverse metric. We also imitate Ivanov's proof of the same fact, using discrete differential forms. And we prove a new theorem: the FAC holds for M¨obiusbands with self- reverse metric. To prove this we use a combinatorial curve shortening flow developed by de Graaf-Schrijver and Hass-Scott. The same method yields a proof of the systolic inequality for Klein bottles with self-reverse metric, conjectured by Sabourau{Yassine. Self-reverse metrics can be discretized using walls because every normed plane satisfies Crofton's formula: the length of every segment equals the symplectic measure of the set of lines that it crosses.
    [Show full text]
  • An Brief Introduction to Finsler Geometry
    An brief introduction to Finsler geometry Matias Dahl July 12, 2006 Abstract This work contains a short introduction to Finsler geometry. Special em- phasis is put on the Legendre transformation that connects Finsler geometry with symplectic geometry. Contents 1 Finsler geometry 3 1.1 Minkowski norms . 4 1.2 Legendre transformation . 6 2 Finsler geometry 9 2.1 The global Legendre transforms . 9 3 Geodesics 15 4 Horizontal and vertical decompositions 19 4.1 Applications . 22 5 Finsler connections 25 5.1 Finsler connections . 25 5.2 Covariant derivative . 26 5.3 Some properties of basic Finsler quantities . 27 6 Curvature 29 7 Symplectic geometry 31 7.1 Symplectic structure on T ∗M n f0g . 33 7.2 Symplectic structure on T M n f0g . 34 1 Table of symbols in Finsler geometry For easy reference, the below table lists the basic symbols in Finsler ge- ometry, their definitions (when short enough to list), and their homogene- ity (see p. 3). However, it should be pointed out that the quantities and notation in Finsler geometry is far from standardized. Some references (e.g. [Run59]) work with normalized quantities. The present work follows [She01b, She01a]. Name Notation Homogeneity (co-)Finsler norm F; H; h 1 1 @2F 2 gij = 2 @yi @yj 0 ij g = inverse of gij 0 1 @gij 1 @3F 2 Cartan tensor Cijk = 2 @yk = 4 @yi @yj @yk −1 i is Cjk = g Csjk −1 @Cijk Cijkl = @yl −2 Geodesic coefficients Gi 2 G i @ i @ Geodesic spray = y @xi − 2G @yi no i @Gi Non-linear connection Nj = @yj 1 δ @ s @ Horizontal basis vectors δxi = @xi − Ni @ys no i i @Nj @2Gi Berwald connection Gjk = @yk = @yj @yk 0 Chern-Rund connection Γijk 0 i is Γjk = g Γsjk 0 Landsberg coefficients Lijk m Curvature coefficients Rijk 0 m Rij 1 m kL −1 Rij = Rijky m 2 i ij Legendre transformation L = h ξj 1 L ∗ L −1 j : T M ! T M i = gijy 1 2 1 Finsler geometry Essentially, a Finsler manifold is a manifold M where each tangent space is equipped with a Minkowski norm, that is, a norm that is not necessarily induced by an inner product.
    [Show full text]
  • On Complex Finsler Manifolds
    ON COMPLEX FINSLER MANIFOLDS 著者 "AIKOU Tadashi" journal or 鹿児島大学理学部紀要. 数学・物理学・化学 publication title volume 24 page range 9-25 別言語のタイトル 複素フィンスラー多様体について URL http://hdl.handle.net/10232/6485 ON COMPLEX FINSLER MANIFOLDS 著者 AIKOU Tadashi journal or 鹿児島大学理学部紀要. 数学・物理学・化学 publication title volume 24 page range 9-25 別言語のタイトル 複素フィンスラー多様体について URL http://hdl.handle.net/10232/00012473 Rep. Fac. Sci., Kagoshima Univ., (Math., Phys. & Chem.), No. 24, p. 9-25, 1991. ON COMPLEX FINSLER MANIFOLDS Dedicated to Professor Dr. Masao Hashiguchi on the occasion of his 60th birthday Tadashi Aikou* (Received September 10, 1991) Abstract The purpose of the present paper is to investigate complex Finsler manifolds and introduce some special complex Finsler structures to consider the analogy of real Finsler geometry. Furthermore, we give a note on the holomorphic sectional curvature of complex Finsler manifolds. Introduction We know already many papers on complex Finsler geometry (cf. Fukui [1], Ichyyo [4, 6], Kobayashi [7], Rizza [12, 13], Royden [14], Rund [15], etc.). Suggested by Kobayashi [7] and Royden [14], in the present paper, we shall investigate complex Finsler manifolds. Inァ1 we shall introduce the notions of complex Finsler vector bundle and complex Fmsler connection, and inァ2, by using the so-called non-linear connection, we show the local expressions of complex Finsler connections. Inァ3 we shall investigate two types of complex Fmsler connections which are determined from the given complex Finsler metric. The first one is the Hermitian connection in a complex Finsler vector bundle with a Hermitian metric, that is, the Finsler-Hermitian connection, which is essentially the same as the one treated in Kobayashi [7].
    [Show full text]
  • On Finsler Manifolds of Negative Flag Curvature
    On Finsler manifolds of negative flag curvature Yong Fang and Patrick Foulon D´epartement de Math´ematiques,Universit´ede Cergy-Pontoise, 2, avenue Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France (e-mail : [email protected]) CIRM, Luminy, case 916, 13288 Marseille Cedex 9, France (e-mail: [email protected]) Abstract One of the key differences between Finsler metrics and Rieman- nian metrics is the non-reversibility, i.e. given two points p and q, the Finsler distance d(p; q) is not necessarily equal to d(q; p). In this pa- per, we build the main tools to investigate the non-reversibility in the context of large-scale geometry of uniform Finsler Cartan-Hadamard manifolds. In the second part of this paper, we use the large-scale geometry to prove the following dynamical theorem: Let ' be the geodesic flow of a closed negatively curved Finsler manifold. If its Anosov splitting is C2, then its cohomological pressure is equal to its Liouville metric entropy. This result generalizes a previous Riemannian result of U. Hamenst¨adt. 1 Introduction Let M be a C1 manifold and TM be its tangent bundle. A C1-Finsler + metric on M is a function F : TM ! R satisfying: (a) F (tu) = tF (u) for any u 2 TM and t ≥ 0; (b) F is strictly positive and C1 over TM − f0g; (c) In standard local coordinates (xi; yi) on TM, the matrix of partial deriva- 2 2 tives @ F is positive definite. @yi@yj A Finsler metric F is said to be Riemannian if there exists a C1- p Riemannian metric g such that F = g.
    [Show full text]
  • Pre-Publication Accepted Manuscript
    Enrico Le Donne, Sebastiano Nicolussi Golo Regularity properties of spheres in homogeneous groups Transactions of the American Mathematical Society DOI: 10.1090/tran/7038 Accepted Manuscript This is a preliminary PDF of the author-produced manuscript that has been peer-reviewed and accepted for publication. It has not been copyedited, proof- read, or finalized by AMS Production staff. Once the accepted manuscript has been copyedited, proofread, and finalized by AMS Production staff, the article will be published in electronic form as a \Recently Published Article" before being placed in an issue. That electronically published article will become the Version of Record. This preliminary version is available to AMS members prior to publication of the Version of Record, and in limited cases it is also made accessible to everyone one year after the publication date of the Version of Record. The Version of Record is accessible to everyone five years after publication in an issue. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000{000 S 0002-9947(XX)0000-0 REGULARITY PROPERTIES OF SPHERES IN HOMOGENEOUS GROUPS ENRICO LE DONNE AND SEBASTIANO NICOLUSSI GOLO Abstract. We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main exam- ples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances.
    [Show full text]